The goal of this clinical research study is to learn if it is safe and feasible to transplant changed cord blood for patients with leukemia or lymphoma. Researchers also want to learn if this can help to control the disease.

The cord blood will be changed to make use of sugar that is found in small amounts in blood cells. It plays a role in signaling where in the body the transplanted cells should go to. Adding more sugars to the cord blood cells in the laboratory is designed to help the cord blood cells find their way faster to the bone marrow. This may help your blood counts to recover faster. This process is called fucosylation.

Anti-thymocyte globulin (ATG) is a protein that removes immune cells that cause damage to the body.

Clofarabine is designed to interfere with the growth and development of cancer cells.

Fludarabine is designed to interfere with the DNA (genetic material) of cancer cells, which may cause the cancer cells to die. This chemotherapy is also designed to block your body's ability to reject the donor's bone marrow cells.

Melphalan and busulfan are designed to bind to the DNA of cells, which may cause cancer cells to die.

Mycophenolate mofetil (MMF) and tacrolimus are designed to block the donor cells from growing and spreading in a way that could cause graft versus host disease (GVHD -- a condition in which transplanted tissue attacks the recipient's body). This may help to prevent GVHD.

Rituximab is designed to attach to cancer cells, which may cause them to die.

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